Theories of the Universe: Planck's Constant

Planck's Constant

Theories of the Universe

In Cracks in a Newtonian World, we briefly covered the ultraviolet catastrophe. If you don't remember exactly what that was, now would be a good time to go back and review it. Max Planck was the man who came up with the solution to it and gave birth to quantum mechanics. Let's look at what he discovered.

Universal Constants

A black body is a hypothetical object that absorbs all electromagnetic radiation that falls on it. Such an object, if heated, would be a perfect radiator, producing black body radiation. Such a hot black body would be no longer black, because it would be radiating visible light. The best example of a black body in a laboratory is a container with a small hole in it into which radiation shines and is trapped. When the container is heated, radiation bounces around inside of it and eventually escapes from the hole as black body radiation.

The ultraviolet catastrophe deals with something called black body radiation. The important point about this type of radiation is that its properties depend only on temperature. If we plot a curve on a graph representing the spectrum of the radiation, it looks like a smooth hill, with a peak at a frequency (color) that depends on its temperature. The hotter the radiation, the higher the frequency at which the peak occurs. This relates directly to the earlier explanation in Cracks in a Newtonian World where the color of an object that was heated slowly changed color from red to white.

Until Max Planck developed the first quantum theory of radiation, the shape of the black body curve was a mystery, since it could not be explained by the classical physics of Maxwell's equation of electromagnetism. The problem was that if electromagnetic waves are treated mathematically in the same way as strings on a violin, and if waves can be any size, classical theory predicts that when energy (heat) is put into any object and radiated as electromagnetic waves, the amount of energy radiated at each frequency is proportional to the frequency. In other words, the higher the frequency, the more radiation there should be. A black body should emit huge amounts of energy in the highest frequency (shortest wavelength) part of the spectrum, in the ultraviolet and beyond, which of course it doesn't.

Universal Constants

Planck's constant is the ratio of a particle's energy to its frequency. Mathematically this is written h = E ÷ f, where h is the symbol for the constant. So, if a particle's frequency increases, its energy must also increase. If its frequency decreases, its energy will decrease as well. But Planck's constant always stays the same and is always equal to one quantum. Another way of putting it is that it relates energy's particle nature to its wave nature. It's represented by the number 0.000 000 000 000 000 000 000 000 006 626, or about a billionth of a billionth of a billionth of 1. But it is not 0. If it were, we wouldn't be able to sit in front of a fire. It's time to be grateful for the little things.

Planck found a way to avoid this problem. He cut up the radiation, mathematically, into chunks, or quanta. At a particular frequency f, each quantum of radiation has an energy E, given by the equation E = hf, where h is a constant in nature, now known as Planck's constant. Let's see how this works.

In any object, the energy is distributed among the atoms according to the temperature of the object. A few atoms have low energy, a few have high energy, and a lot have the middle amount of energy. This large amount of middle energy atoms increases as the temperature increases. Each atom can emit electromagnetic radiation. For very high frequencies (large values of f), the energy needed to emit one quantum of energy (e) is very large, and only a few atoms in the black body will have that much energy available, so only a few high-frequency quanta are radiated. At very low frequencies, it is easy for atoms to emit low-energy quanta, but they have so little energy that even added together they do not amount to much. In between the two extremes, however, there are many atoms that have enough energy to emit moderate-sized quanta of radiation. These add up to produce the peak in the black body curve. And the peak shifts to higher frequencies for hotter bodies, because in hotter bodies there are more individual atoms with greater amounts of energy. And that's it!